[code0]

As you said, "average the overall probabilities between a regular game and a switch-style game", which is 1/3 for a game with no-switching and 2/3 for a game with always-switching. Averaging gives you a probability of 1/2. So if you switch, you win 50 % of the time. But so do you if you do not switch. So there is no benefit in switching, which was basically my point.

EDIT: In both the versions of the game, I am assuming an omniscience host who always opens the door with a goat behind, if he does open a door at all.

[tel]

You're absolutely right and my argument fails, but it doesn't invalidate the point I wanted to get at. In the long run, your coin flipping method does cause the game to behave as though there were only two doors, but in any given game you'll know whether or not the host is going to choose a door.

That means that, conditional on a good flip, you may still be able to act to your advantage since there's information transfer from the host to you.